Experimental Determination of Pipe and Pipe Fitting Losses

Transcription

1 Experimental Determination of Pipe and Pipe Fitting Losses ME333, Introduction to Fluid Mechanics Date Performed: 11 March 2010 Date Submitted: 19 March 2010 Group Members: Chris Chapman Joy Ann Markham Sean Elliott Stoker Trevor Crain 1

2 Table of Contents Executive Summary:... 3 A. Introduction... 4 h L major... 5 h L minor... 5 B. Procedure... 6 C. Results... 7 C.1. Pipe A: 90 Degree Elbow Fitting... 7 C.2. Pipe A: T- Bend Fitting... 8 C.3. Combined Results D. Discussion Appendix A: Raw Data and Measurements Appendix B: Normalized and Converted Data Appendix D: Determination of Roughness, ε Appendix E: Moody Chart Appendix F: Determination of K L values Appendix G: References Appendix I: Individual Statements of Contribution

3 Executive Summary: The goal of this experiment was to determine the loss coefficients for pipe system components and the roughness values for straight pipes using experimental data. The effects of Reynolds number on the friction factors in straight pipes were also investigated. To perform this experiment, a pipe system consisting of two pipe segments connected perpendicular to one another with a 90 elbow component or a T-Bend fitting was used. Air was pumped through the pipes at a designated flow rate and the pressure readings along the pipes were recorded using taps connected to a digital manometer. The flow rates were determined from a set of three Reynolds numbers to be tested. For this experiment the Reynolds numbers of 15000, 25000, and were used to determine the flow, all of which were considered turbulent. From the pressure values obtained at these three Reynolds numbers the friction factor and roughness for the straight pipe as well as the loss coefficient for the bend fittings were calculated. The average friction factors for the 90 elbow and T-Bend configurations were and respectively. The 90 elbow fitting was found to have an average loss coefficient of and the T-Bend was found to have a coefficient of This indicated that the 90 elbow component is a better choice for use in a 90 angle setup as it had the lowest loss coefficient. The resulting friction factors at different Reynolds numbers did not conclusively show how these values change with Reynolds numbers. The values did not generally increase or decrease with an increase in Reynolds number; according to the Moody chart, however, they should have decreased with increasing Reynolds numbers. For the roughness, measured values were obtained using equation D.1 and theoretical values were gathered from the Moody chart. Error calculations showed that the results for roughness were acceptably accurate with maximum errors of only 13% for the T-Bend. The data also showed that the higher the Reynolds number, correspond to more accurate the roughness calculations, with an error of only 2.5% for the T- Bend at a Reynolds number of

4 A. Introduction Turbulent pipe flow tends to be very difficult to quantitatively understand from a purely theoretical standpoint, so experimental analysis is necessary to place some solid numerical backing behind this theory. This lab was intended to explore the effects of various flow rates (and consequently velocity and Reynolds number) on pressure drops through a fixed diameter pipe system with replaceable joints. This system is illustrated in Figure 1. Figure 1: Pipe system Within a pipe system, there are two types of losses. The first is a Major Loss, and consists of the head losses due to viscous effects in straight segments of pipe in the system. This will be hereafter referred to as h L major. The second type is a Minor Loss, and is comprised of losses generated within components of the pipe system other than the straight pipes themselves. This will be hereafter referred to as h L minor. 4

5 h L major The viscous effects mentioned earlier are a result of the shear stresses that exist due to interaction between the pipe wall and the fluid flow. The pressure drop through these systems is known to be dependent of a number of other parameters, such that:! =!(!,!,!,!,!,!) Here V is average velocity, D is the pipe diameter, l is the length of pipe being considered, ε is the roughness of the given pipe, µ is the dynamic viscosity, and ρ is the fluid density. Two other important quantities for understanding turbulent pipe flow are the Reynolds Number, hereafter denoted Re, and the roughness to diameter ratio, known as relative roughness, and calculated as ε/d. These quantities are used in conjunction with a Moody Chart (Figure E.1) to generate a true value for the friction factor, f, which is ultimately used to calculate the h L major. Once the true value for the friction factor has been obtained, the calculation for h L major is:!! h!!"#$% =!!! 2! h L minor Minor losses are not necessarily minor in relation to major losses. Depending on the exact configuration of the pipe system in question, they can actually be more significant than the major losses. These minor losses are generated in components such as elbows, 180-degree bends, tees, valves, and reducers. As the calculation of h L minor is difficult through traditional means, it is best to analyze the losses through these elements by computing an equivalent straight pipe length, denoted K L, which simulates the same reduction in flow energy as the complex element would. This value is experimentally obtained, and is then used in the calculation of the head loss, h L minor, through the given element. See Appendix A for detailed values of K L. Once the correct value for K L is selected, it can be used in a formula similar to that used for major losses, specifically: h!!"#$% =!!!! 2! 5

6 B. Procedure 1. Initial Measurements and calculations 1.1. Read the barometric pressure prior to conducting experiment 1.2. Determine the temperature of the ambient air 1.3. Record each of the dimensions listed in Figure Using calipers, measure the ID of the pipe system, and calculate the area from this value 1.5. Calculate the heights, h, on a manometer necessary to achieve a Re value of 15000, 25000, and through the pipe system. 2. Setup 2.1. Activate the manometer 1 hour prior to testing 2.2. position the yellow dump valve so that the handle is parallel to the pipe it is attached to (this is fully open) 2.3. Press the green start button to begin running the supply fan and air conditioner 2.4. Calibrate the thermostat at the flow bench to match the temperature recorded previously 2.5. Set the multipoint selector switch set to position 1, and verify that the temperature displayed on the thermostat still agrees with the ambient reading. 3. Calibration of Instruments and Pipe Flow rate 3.1. To zero the manometer: Set the three position filter switch to OFF Set the scale knob to position X Use the zeroing knob to adjust manometer until it reads Move three position filter switch from OFF to HI 3.2. Adjusting flow rate: position the yellow dump valve so that the handle is parallel to the pipe it is attached to (this is fully open) Verify that the valve is open by placing a hand in front of the opening following the valve, and noting the presence of airflow Use the blue dial valve to adjust the flow within the system so that the height, h, on the manometer reads the appropriate value calculated in step 1.5. Note that if the correct reading cannot be achieved, slowly close the yellow dump valve until the reading on the manometer is above the desired reading, then bring back down using the blue dial valve 4. Pipe A: 90 Degree Elbow Fitting 4.1. Adjust flow until the manometer reads the height that corresponds to Re = 15000, using the techniques listed in steps Set the knob labeled HP, the high pressure source, to PIPE A 4.3. Set the knob labeled LP, the low pressure source, to R1 6

7 4.4. Verify that the position of the handle on the ball valve for pipe A is parallel to the pipe (fully open), and that the handle on pipe B is perpendicular to the pipe (fully closed) 4.5. Set the selector knob for Pipe A to position Record the differential pressure displayed on the electronic manometer Repeat steps 4.5 and 4.6 for ports Pipe A: 90 Degree Tee Fitting 5.1. Replace the 90 degree fitting the with the capped Tee fitting. Verify that the joints are properly sealed by feeling for any airflow around the joints 5.2. Repeat steps for this fitting. 6. Shut Down 6.1. Press the red stop button for the fan and air conditioner unit 6.2. Close the blue dial valve completely C. Results The raw data and Reynolds number calculations are listed in Appendix A. The pressure at each tap along with the geometry of the test setup was used to analyze the flow of air through the pipe and 90 degree elbow and T bends. C.1. Pipe A: 90 Degree Elbow Fitting The normalized raw data found in Appendix B was used to create a plot of pressure versus length along pipe seen in Figure 2. Figure 2. Plot of Pressure versus length along pipe for 90-degree elbow setup. 7

8 The friction factor for each Reynolds number and average friction factor are listed in Table 1. Details of the calculations of the friction factor are contained in Appendix C. Table 1. Friction factors for the 90 degree elbow bend setup Reynolds Number Friction Factor, ƒ Average Table 2 contains values for the theoretical and measured relative roughness and roughness for each Reynolds number and the percent error between the theoretical and measured roughness values. Appendix D contains details of the roughness calculations. Table 2. Theoretical and measured values of ε/d and ε for the 90 degree elbow bend Reynolds Number Estimated ε/d Calculated ε/d Theoretical ε (mm) Measured ε (mm) Percent Error (%) (Smooth) (Smooth) The calculated values of the loss coefficient for the 90 degree elbow bend are contained in Table 3. Details of these calculations can be seen in Appendix E. Table 3. Loss coefficient values for the 90 degree elbow bend Reynolds Number Measured Loss Coefficient, K L Average C.2. Pipe A: T-Bend Fitting Figure 3 displays a plot of pressure versus length along pipe for the T-bend setup. Data for this plot can be found in Appendix B. 8

9 Figure 3. Plot of Pressure versus length along pipe for T bend setup. The friction factor for each Reynolds number and average friction factor are listed in Table 4. Details of the calculations of the friction factor are contained in Appendix C. Table 4. Friction factors for the T bend setup Reynolds Number Friction Factor, ƒ Average Table 5 contains values for the theoretical and measured relative roughness and roughness for each Reynolds number and the percent error between the theoretical and measured roughness values. Appendix D contains details of the roughness calculations. Reynolds Number Table 5. Theoretical and measured values of ε/d and ε for the T bend setup Estimated ε/d Calculated ε/d Theoretical ε (mm) Measured ε (mm) Percent Error (%)

10 The calculated values of the loss coefficient for the T bend are contained in Table 6. Details of these calculations can be seen in Appendix E. Table 6. Loss coefficient values for the T bend Reynolds Number Measured Loss Coefficient, K L Average C.3. Combined Results The combined results from both 90 degree elbow and T bend setups are listed in Table 7. These results include the overall average friction factor for the pipe, the range of roughness values, and the average loss coefficients, K L, for each pipe fitting. Table 7. Average ƒ and ε of pipe and K L of each pipe fitting Component Friction Factor, ƒ Roughness, ε (mm) Loss Coefficient Pipe T-Bend Degree Elbow D. Discussion The goals of this lab were to determine the effects of Reynolds number on the friction factor for straight pipes and to find the loss coefficients for both a 90 elbow and T-bend fitting. The roughness for the straight pipe was also calculated. The calculated results for the roughness were fairly accurate with the results obtained from the literature s Moody Chart, with errors less than 13%. The results were inconclusive as to what effect increasing the Reynolds number had on the friction factor. According to the Moody chart, the friction factor should decrease with increasing Reynolds numbers; however, the calculated friction values from the experiment increased from to and then decreased from to This discrepancy from the behavior predicted in the literature was most likely due to the many sources of error present in the experiment. One possible source of error came from the pressure readings. The pressure readings at the reference point for each component and each flow was some value greater than zero (Tables A.5 and A.6), but the problem was that all the reference point readings should have been zero regardless of the set up. The reason for this discrepancy remains undetermined, however it is suspected that there was a problem with the machine s manometer. This theory was supported by the fact that when taking readings at the various reference taps, the manometer value never 10

11 stabilized; instead, it would often slowly decrease, increase, or bounce around randomly. Readings were taken when the pressure value shown appeared the most stable, however no reading was ever truly stable. Despite this problem, the data obtained was still used, and to account for the reference pressure not being zero, the data was normalized (Table B.1 and B.2). The ΔP values obtained from the data should still be the same, with or without normalizing, so the high reference pressure should not affect the friction factors calculated. The real problem with having incorrect reference pressures was the aforementioned implication that the manometer took incorrect readings. That combined with the unstable pressure readings is where the error would come from. It is hard to say precisely in what matter or how significantly these incorrect readings affected the results and the values calculated from the results since the pressure readings were not consistently incorrect. For instance, when the pressure readings constantly decreased at one reference tap, then the overall ΔP was probably higher than it should have been for that section, but if the pressure readings were constantly increasing, then the ΔP would be smaller than it should have been, making the friction factors higher and lower for each case respectively. It is likely that this problem caused the bulk of the error of the data, perhaps causing as much as 50% of the error for the lab. Another source of error came from the connections between the two different branch components, the T-bend and the 90 elbow. When components were changed it was necessary to attempt to perfectly line up both Pipe A and the pipe segment after the branch component with the openings of the branch component. However, these connections did not line up perfectly every time, which may have caused small air leakages at the connections. These small leaks would lead to a greater pressure drop across the component than the ideal situation. According to equation F1, K L is proportional to ΔP, which means that the increased ΔP across the elbow or T- bend would have resulted in a K L value that was higher than expected. However, the connections were inspected before each test to verify that no air could be felt escaping from the pipes. This ensured that any leaks from the connection points were very small, so it is possible that this error source did not contribute to more than 30% of the errors in our K L values. The misaligned branch component would also have caused an increase in ΔP because parts of the pipe ends would be jutting into the airstream at the connection points. These protrusions into the airstream would have obstructed some of the flow, leading to an increased drop in pressure across the branch component. As stated above, this greater value for ΔP would have resulted in a higher experimental K L value. It is difficult to ascertain whether this source of error or the leakages contributed more to the higher-pressure drops measured, though due to the fact that no obvious leaks were observed it can be predicted that a greater amount of the error in the K L results was due to the misalignment of pipes. 11

12 Appendix A: Raw Data and Measurements Figure A.1 displays a schematic of the experimental setup. Figure A.1. Schematic of the experimental piping setup. Table A.1 contains the measurements of the pipe setup as defined in Figure 1. Table A.1. Measurements of pipe setup Interval Length (mm) L 1 25 L L L L 5 25 L L U 253 L D 251 Tap Spacing 10 Pipe Diameter, D (mm) 26.5 Table A.2 contains the measurements of atmospheric pressure and temperature at the time of the experiment. 12

13 Atmospheric Pressure (mm Hg) Table A.2. Atmospheric pressure and temperature Atmospheric Pressure (kpa) Ambient Temperature ( F) Ambient Temperature (K) Data was to be taken for flow with Reynolds values of 15,000, 25,000, and 35,000. In order to determine the flow rates necessary to produce these values, the cross-sectional area of the pipe and the necessary air velocity was calculated by Equations A.1 and A.2 respectively. A.1 In these equations D is the pipe diameter, Re is the desired Reynolds number, and υ is the kinematic viscosity of air determined by Equation A.3, known as the Sutherland Equation, and Equation A.4. A.2 A.3 In these equations µ is the dynamic viscosity of air, T is the temperature in Kelvin, ρ is the density of air, and C and S are constants. C and S were determined from known values of µ at certain temperatures to be: A.4 The density of the air was determined by the ideal gas equation in Equation A.5. In this equation P is the atmospheric pressure in Pascals, T is the temperature in Kelvin, and R is the gas constant of air. Table A.3 contains the value of R along with the density and kinematic viscosity of air at the temperature and pressure the experiment was performed at. Table A.3. Gas constant and Kinematic and dynamic viscosities of air R (J/(kg*K)) Kinematic Viscosity, ν (m 2 /s) Density, ρ (kg/m 3 ) * A.5 13

14 Once the values of velocity and area were determined, the flow rate, Q, was found from Equation A.6. A.6 The required flow meter pressure, measured in millimeters of mercury, was then determined from the relationship in Equation A.7. A.7 The values for velocity, flow rate, and flow meter pressure required for each Reynolds number are contained in Table A.4. Table A.4. Velocities, flow rates, and pressure for each Reynolds number Reynolds Number Velocity (m/s) Q (m 3 /s) H (mm Hg) Tables A.5 and A.6 contain the raw data as recorded in the lab for the flow with each Reynolds number for the 90 degree elbow bend and T bend setup respectively. Reynolds Number Flow Meter, H (mm H 2 O) Table A.5. Raw data for the 90 degree elbow bend setup Tap Number Relative Pressure (mm Hg)

15 Table A.6. Raw data for the T bend setup Reynolds Number Flow Meter, H (mm H 2 O) Tap Number Relative Pressure (mm Hg)

16 Appendix B: Normalized and Converted Data In order to more easily analyze the raw data, the data was normalized so that the pressure reading at each tap was taken with respect to a zero value of pressure at the eleventh tap. The pressure at each tap was also converted to Pascals using Equation B.1 and the flow meter pressure to flow rate, Q, using Equation A.7. Tables B.1 and B.2 contain the normalized raw data and the distance from tap 1 of each following tap for each experimental setup. B.1 In this equation P Pa is the pressure in Pascals and P mmhg is the pressure in millimeters of mercury. Table B.1.Normalized and converted data for the 90 degree elbow bend setup Reynolds Number Flow Rate, Q (m 3 /s) Tap Number Distance (m) Relative Pressure (Pa)

17 Table B.2.Normalized and converted data for the T bend setup Reynolds Number Flow Meter (m 3 /s) Tap Number Distance (m) Relative Pressure (Pa)

18 Appendix C: Determination of the Friction Factor, ƒ The friction factor of the pipe was determined by performing a least squares approximation on the normalized data. Equation C.1 describes the method of least squares approximation. C.1 In this equation m is the slope of the line of best fit through N data points of pressure, P, versus length, l. Plots of pressure versus length were created for each Reynolds number of each setup using Microsoft Excel. Lines of best fit were inserted for taps 1 through 6 and taps 7 through 11 in order to eliminate the effect of the T bend or 90 degree elbow. The plots and the equations for each line are displayed in Figures C.1 and C.2. Figure C.1.Plots of pressure versus length for the 90 degree elbow bend setup. 18

19 Figure C.2. Plots of pressure versus length for the 90 degree elbow bend setup. In this experiment the pipe flow between taps other than 6 and 7 (where the bend was located) was modeled by Equation C.2. The slope of an equation of a line of best fit corresponds to a value of since the plots are of pressure versus length. Values of the friction factor, ƒ, were found by taking the average of the slopes from each segment (taps 1 through 6 and taps 1 through 7) for each Reynolds number and then dividing by (-ρv 2 )/(2D), where ρ, V, and D are all known values. The average of theses values was then taken to find the average friction factor. Table C.1 and C.2 contain values for the slope of each line segment, the average slopes, the values of (ρv 2 )/(2D), and the friction factors for each Reynolds number for each setup as well as the average friction factor. 19 C.2

20 Reynolds Number Table C.1. Friction factors for 90 degree elbow bend setup Slope of taps 1-6 Slope of taps 7-11 Average slope, m avg (-ρv 2 )/(2D) (N/m 3 ) ƒ Average ƒ Reynolds Number Slope of taps 1-6 Table C.2. Friction factors for T bend setup Slope of taps 7-11 Average slope (ρv 2 )/(2D) (N/m 3 ) ƒ Average ƒ The overall average friction factor was found by averaging the average friction value from each setup. This value was determined to be: ƒ =

21 Appendix D: Determination of Roughness, ε The measured values of relative roughness, ε/d, and roughness, ε, for each Reynolds number of each test setup was determined from the relationship between the friction factor and the Reynolds number for turbulent flow (Re > 2100). Equation D.1 defines this relationship. D.1 The measured values of ƒ for each Reynolds number of each setup were used in this equation to calculate ε/d for each Reynolds number of each setup. The Moody Chart in Appendix E was used to find theoretical values of ε/d for each Reynolds number and corresponding measured ƒ value for each setup. The value for each Reynolds number of each setup is marked on the Moody Chart. The theoretical and measured values of ε/d were multiplied by D to obtain values of roughness. Tables D.1 and D.2 contain the theoretical and measured values of ε/d and ε and the percent error, as determined by Equation D.2, between the measured and theoretical values of ε for each setup. D.2 Reynolds Number Table D.1. Measured and Theoretical values of ε/d and ε for 90 degree elbow bend Theoretical Relative Roughness, ε/d Measured Relative Roughness, ε/d Theoretical Roughness, ε (mm) Measured Roughness, ε (mm) Percent Error (%) (Smooth) (Smooth) Reynolds Number Table D.1. Measured and Theoretical values of ε/d and ε for T bend Theoretical Relative Roughness, ε/d Measured Relative Roughness, ε/d Theoretical Roughness, ε (mm) Measured Roughness, ε (mm) Percent Error (%)

22 Appendix E: Moody Chart Figure E.1 contains a Moody chart with experimental values marked on it used to determine the theoretical values of relative roughness. Figure E.1. Moody Chart to determine theoretical relative roughness Source: 22

23 Appendix F: Determination of K L values The values of K L for the 90 degree elbow bend and T bend were determined from the relationship between K L and pressure difference, ΔP, as defined in Equation F.2. Here ΔP is equal to the pressure before the component minus the pressure after the component. The pressure before the component was found by taking the pressure at tap 6 and subtracting the head loss between it and the beginning of the bend. The head loss is equal ƒlρv 2 /(2D). This is found taking the average slope of the two segments in the pressure versus length graph for the desired Reynolds number and multiplying it by the length of L U. This operation is defined in Equation F.2. F.2 The pressure after the bend was found in a similar manner except that the head loss was added instead of subtracted since the pressure at tap 7 is lower than the pressure at the end of the bend. The determination of the pressure after the bend is detailed by Equation F.3. F.3 Tables F.1 and F.2 contain the values of P Before, P after, ΔP, 2/(ρV 2 ), and K L for each Reynolds number of each setup as well as the average values for K L. Table F.1. Pressure drop and Loss Coefficient values, K L, for 90 degree elbow bend Reynolds Number P Before (Pa) P After (Pa) ΔP (Pa) 2/(ρV 2 ) K L F.1 Average K L 0.74 Reynolds Number Table F.2. Pressure drop and Loss Coefficient values, K L, for T bend P Before (Pa) P After (Pa) ΔP (Pa) 2/(ρV 2 ) K L Average K L

24 Appendix G: References Munson, Bruce Roy, Donald F. Young, and T. H. Okiishi. Fundamentals of Fluid Mechanics. Hoboken, NJ: J. Wiley & Sons,

25 Appendix I: Individual Statements of Contribution Sean Stoker: I was responsible for the calculations, results, and appendices of the report. I also helped in collectively editing and putting the report together. Signature Date Chris Chapman: -Cover Page -Introduction -Procedure -Editing Signature Date Joy Markham -Executive Summary -Discussion -Formatting and Editing Signature Date Trevor -Executive Summary -Discussion -Formatting and Editing Signature Date 25